Ogilvie 



pressure is not integrable, and so one must use the near-field ex- 

 pansion for any force calculation. Furthermore, it is a disturbing 

 result, because it suggests that many previous attempts to incor- 

 porate bow-wave nonlinearities into linear-theory singularities 

 have been futile exercises. 



Personally, I am not yet willing to admit that the possibility 

 of having the complex velocity behave like Z' '' is really to be re- 

 jected, as Dagan and Tulin claim. Wagner [ 193 2] analyzed the 

 region of the jet and the stagnation point for the flow against a flat 

 plate of infinite extent downstream, and he showed that this flow, 

 from far away, has the behavior of a flow around the leading edge 

 of an airfoil, that is, the velocity varied with Z' , Physically it 

 seems rather difficult to imagine that, by curving the body around 

 just behind the stagnation point, one causes such a drastic change in 

 the apparent singularity. 



Dagan and Tulin present a figure (their Fig, 2) in which they 

 have placed many symbols showing beam/draft ratios of more than 

 a hundred ships, and it is quite evident that most ships have values 

 of this ratio considerably greater than unity. They then use this fact 

 as an alleged justification for claiming that their 2-D model of the 

 bow flow (as in Fig. (5-5)) will have some validity in describing the 

 flow around the bow of an actual ship -- since most ships are pre- 

 sumably of the "flat" variety. However, this claim is completely 

 misleading. The theory might apply to a scow, but not to a ship. 

 After all, beam/draft ratio is measured amidships, and even ships 

 with the largest block coefficients have entrance angles less than 

 180O. 



Also, it is appropriate to mention again the warning against 

 defining a small parameter precisely and then trying to interpret on 

 some absolute basis whether a particular value of the parameter is 

 "small enough. " For example, it is conceivable that a thin-ship 

 analysis would be valid for a ship with beam/draft ratio of 10, 

 whereas a flat- ship analysis might fail for the same ship. 1 am not 

 saying that this is likely, but it is possible. In one problem, a 

 value of 10 might be "small," whereas in another problem a value 

 of 1/10 might be "not small. " 



Notwithstanding these objections, the paper by Dagan and 

 Tulin has provided a refreshing change in outlook on the bow-flow 

 problem, and perhaps it will be more fruitful eventually than the 

 usual attempts to place complicated singularities at the bow in the 

 frame-work of linearized theory. 



786 



